We study a new family of measures of noncompactness in the locally Sobolev space ,1 loc (R + ), equipped with a suitable topology. As an application of the technique associated with this family of measures of noncompactness, we study the existence of solutions for a class of nonlinear Volterra integrodifferential equations. Further, we give an illustrative example to verify the effectiveness and applicability of our results.

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In this paper, first we describe the notion of dislocated S b-metric space and then we introduce the new notions of generalized F-contraction and generalized F-Suzuki-contraction in the setup of dislocated S b-metric spaces. We establish some fixed point theorems involving these contractions in complete dislocated S b-metric spaces. We also furnish some examples to verify the effectiveness and applicability of our results.